After Googling through geneological records dealing with the Helm and Taliaferro families, it seems that Linai is probably a respelling (adopted only by this particular individual) of Lynaugh or Lynagh, a surname which became a given name in the Helm and Taliaferro families after a Margaret Lynaugh married Thomas Helm in Virginia in the early 18th century. According to MacLysaght's The Surnames of Ireland, Lynagh is a surname which comes from Gaelic Laighneach, "the Leinsterman".